Int. J. Appl. Math. Comput. Sci., 2012, Vol. 22, No. 1, 41–53
DOI: 10.2478/v10006-012-0003-z
SIGNED DIRECTED GRAPH BASED MODELING AND ITS VALIDATION FROM
PROCESS KNOWLEDGE AND PROCESS DATA
FAN YANG ∗,∗∗ , S IRISH L. SHAH ∗ , D EYUN XIAO ∗∗
∗
Department of Chemical and Materials Engineering
University of Alberta, 7th Floor, ECERF, 9107 116 Street, Edmonton, AB T6G 2G6, Canada
e-mail: {fyang.cme,Sirish.Shah}@ualberta.ca
∗∗
Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Automation
Tsinghua University, 1 Tsinghuayuan, Haidian District, Beijing 100084, China
e-mail: {yangfan,xiaody}@tsinghua.edu.cn
This paper is concerned with the fusion of information from process data and process connectivity and its subsequent use
in fault diagnosis and process hazard assessment. The Signed Directed Graph (SDG), as a graphical model for capturing
process topology and connectivity to show the causal relationships between process variables by material and information
paths, has been widely used in root cause and hazard propagation analysis. An SDG is usually built based on process
knowledge as described by piping and instrumentation diagrams. This is a complex and experience-dependent task, and
therefore the resulting SDG should be validated by process data before being used for analysis. This paper introduces
two validation methods. One is based on cross-correlation analysis of process data with assumed time delays, while the
other is based on transfer entropy, where the correlation coefficient between two variables or the information transfer from
one variable to another can be computed to validate the corresponding paths in SDGs. In addition to this, the relationship
captured by data-based methods should also be validated by process knowledge to confirm its causality. This knowledge
can be realized by checking the reachability or the influence of one variable on another based on the corresponding SDG
which is the basis of causality. A case study of an industrial process is presented to illustrate the application of the proposed
methods.
Keywords: signed directed graph, transfer entropy, process topology, fault diagnosis, process hazard assessment.
1. Introduction
The scale and complexity of process operations have been
increasing to meet the business and regulatory compliance demands of industry. These industrial processes, however, are often subject to low productivity due to system
faults and/or even hazardous operating conditions because
of abnormal conditions such as faulty operations, equipment degradation, external disturbances, and control system failures. Compared with the traditional fault detection
in local systems, fault detection in large-scale complex
systems is particularly difficult because of the high degree
of interconnections in such systems. A simple fault in such
systems may easily propagate throughout the plant. Such
a type of analysis is similar to hazard operability analysis,
which is initially carried out in a qualitative way followed
by quantitative analysis. It is important to remember that
simple local faults can propagate and spread to a wider do-
main because of the connectivity between units and the topology of the system. Therefore, in root cause analysis of
process faults, it is important to capture connectivity and
topology to use this information to relate to the dynamics
of the systems as captured in process data. This fusion
of information can be further utilized to find the probable
root causes and resulting consequences according to the
current symptom(s). Therefore, the problems of fault detection and propagation require analysis on two fronts: (i)
capturing process connectivity and causality, and (ii) validation of such models using both process knowledge and
process data.
There are several models to describe process topology and cause/effect relations. Causal graphs describe
the causal relations between variables, and they can be
extended to AND/OR causal graphs to describe logical
causal relations (Lig˛eza and Kościelny, 2008). A potential conflict structure, as a subgraph of a causal graph,
F. Yang et al.
42
describes specifically the computational causal relations
used for consistency-based diagnosis (Lig˛eza, 1996). There are also more complex models that describe essentially causal relations. Bond graphs and their extension,
temporal causal graphs (Mosterman and Biswas, 1999),
use different symbols to further describe dynamic characteristics. More precisely, qualitative transfer functions
(Leyval et al., 1994) and differential equations (Montmain
and Gentil, 2000) have been integrated into causal graphs, and complex algorithms are introduced to improve
their correctness (Cheng et al., 2011). Similar or improved approaches were investigated by many researchers,
such as Pastor et al. (2000), Alonso et al. (2003), Fagarasan et al. (2004), and Jan et al. (2007). All of these improved methods depend on mathematical models. If
Matlab/Simulink models of dynamic systems are available, then the diagnostic models can be generated directly
(Górny, 2001). The dependency on models, however, limits the utility in large-scale complex systems.
As a qualitative graphical model, the Signed Directed Graph (SDG) technique has been used to describe
the topology and causality of industrial processes, including both material flow and information flow paths (Iri
et al., 1979). Compared to classic causal graphs, SDGs
add signs to nodes and arcs to include more information
and yet remain simple and qualitative. SDGs can help
us understand the internal relationships between variables
and thus be implemented in hazard assessment (to find the
possible consequence of a fault) and also in root cause
diagnosis (Yang et al., 2010a).
Although the modeling of SDGs can be based on mathematical equations (Maurya et al., 2003a; 2003b; 2004),
it can also be undertaken practically from process knowledge that is readily available from Piping and Instrumentation Diagrams (P&IDs). Although simple, SDGs are
able to capture fairly precise process connectivity information from the direct relationship between variables, as
based on causality. Some techniques to capture connectivity from P&IDs have been developed by Fedai and Drath
(2005), Yim et al. (2006), and Thambirajah et al. (2009)
based on the standard data format of Computer Aided Engineering Exchange (CAEX), though there may also be a
lot of irrelevant information in these files. The SDGs constructed in this way should be validated by the process
data to confirm their correctness.
When dealing with process data, various pairwise
causality capture methods have been proposed (Bauer
et al., 2007; Bauer and Thornhill, 2008), whereas constructing a causal network or determining the topology
purely from data is difficult, if not impossible, unless one
also resorts to process knowledge. In multivariate cases,
knowledge discovery in databases has been studied and
implemented in the DiaSter system (Korbicz and Kościelny, 2010), including the discovery of qualitative and quantitative dependencies. However, when the scale of a pro-
cess is large, most of the data-based methods are essentially pattern recognition, which is difficult to be combined with knowledge-based or model-based methods. Moreover, by these data-based methods, one is likely to run
into a lot of redundancy and errors.
In general, the combination of knowledge and data
reinforces the trust in SDG models and leads to their use
with a high degree of confidence. In this paper, SDG models are used because they can be built by knowledge as
well as data and therefore provide a common description
that facilitates their validation.
It should be noted that verification and validation are
two important steps to be followed before the model can
be used. Palmer and Chung (1999) discussed verification
to ensure that the internal structure of the model is complete, correct, and consistent; this is based on the model
itself. However, the model should also be tested in the
environment in which it is going to be used; in practice,
historical data are employed for this test, which is the purpose of this paper.
This paper is organized as follows. Section 2 provides an overview of the SDG modeling and inference method based on process knowledge. Section 3 introduces
two data-based methods, time delayed cross-correlation
and transfer entropy, to capture causality between two variables. Section 4 describes mutual validation of knowledge description and a data-based causal network. In Section 5 the above methods are evaluated by application to
a real industrial process where we build an SDG and validate it by process data, followed by concluding remarks
in Section 6.
2. SDG modeling by process knowledge
A process variable (or another continuous variable such
as a manipulated one) is represented by a node in a SDG,
and the cause/effect relationship between two variables is
represented by a directed arc between the corresponding
nodes (Iri et al., 1979). The source node means the cause or parent, and the destination node means the effect or
child. For example, an arc from node A to node B implies
that deviation or perturbation in A may cause deviation in
B. A sign ‘+’, ‘−’, or ‘0’ is assigned to a node in comparison with thresholds to denote higher than, lower than, or
within the normal operating region, respectively. Positive
or negative influence between nodes is distinguished by
the sign ‘+’ (promotion) or ‘−’ (suppression) of the arc,
shown as a solid or dotted arc respectively.
The construction procedure of SDGs can be accomplished based on mathematical models, i.e., Differential
and Algebraic Equations (DAEs). For example, the following Differential Equation (DE):
dxi
= fi (x1 , . . . , xn )
dt
(1)
Signed directed graph based modeling and its validation from process knowledge and process data
can be transformed into a set of arcs with the sign
∂fi sgn(xj → xi ) = sgn
,
∂xj x0 ,...,x0
F1
(2)
n
1
where (x01 , . . . , x0n ) denotes the steady state. Algebraic
Equations (AEs) can be transformed into arcs in much the
same way.
In most cases, the SDG is established by qualitative
process knowledge and experience because precise equations are usually unavailable and unnecessary. Nevertheless, process knowledge is available in P&IDs, and it is
suggested that one should capture process topology from
this information, where upstream variables influence downstream variables, external environmental variables influence internal state variables, and manipulated variables
influence controlled variables in a control system.
2.1. SDG modeling within a process unit. For a single unit, several physical quantities reflect the characteristics of the process. The following three types of relationships between these variables can be described as DAEs: DEs, which reflect dynamic causal relationships (Type I), AEs with causal meaning, which include driving force equations, functional relationships, and other algebraic
equalities (Type II), and AEs with no causal relationships
(Type III) (Yang et al., 2010b). For the first two types,
Maurya et al. (2003a) summarized the modeling methods.
The third type can be used as redundant constraints to remove irrelevant solutions (Oyeleye and Kramer, 1988).
Based on these methods, an SDG of a unit can be obtained from qualitative knowledge instead of DAEs. For
example, consider a tank process as shown in Fig. 1, where L is the level in the tank, K is the valve position in
the outlet pipe, F1 and F2 are inlet and outlet flowrates,
respectively. The DAEs of this process are given below:
A
dL
= F1 − F2 ,
dt
√
F2 = K L,
K = αL,
(3)
(4)
(5)
where A is the cross sectional area of the tank and α is the
proportional coefficient of the control law. By these DAEs, an SDG can be built as shown in Fig. 2, where 2(a)
F1
L
K
F2
Fig. 1. Scheme of a tank process.
L
(a)
F2
L
F2
(b)
K
43
F1
F2
L
K
(c)
Fig. 2. SDGs of the tank process: SDG obtained from Eqn.
(3) (a), SDG obtained from Eqns. (4) and (5) (b), combination of (a) and (b) (c).
is obtained from Eqn. (3) (Type I), 2(b) from Eqns. (4)
and (5) (Type II), and 2(c) is a combination of them.
This SDG can also be obtained from process knowledge
that the level is determined by the inlet and outlet and the
outlet flow rate is affected by the level and the valve position. Irrespective of the type of control (P, PI, or PID),
it is shown as an arc from L to K, and it is unnecessary
to write the corresponding DAEs for this case. Accurate
parameters and functions are not important for this usage.
In the process industry, standard units, such as tanks
and exchangers, can be modeled and saved as templates or
modules for reuse. One can build SDGs for special units
by using the qualitative information obtained from process knowledge as well as first-principles or mathematical
models. When all the units are modeled, they should be
connected according to the linkage information.
2.2. SDG modeling between process units. Units are
connected because of material and signal flow paths. The
directions of arcs are consistent with the transportation paths. The arcs should be connected to corresponding variables in different units.
An efficient way of showing these transportations is
via P&IDs in which both material flow and information
flow (control signal) are shown. SDGs can be built by
unfolding the units in P&IDs as unit SDGs and connecting them according to flow directions. Thus SDGs bring
out not only the connectivity and topology between the
units but also the causality between variables.
This plant topology information can be expressed by
a text file that can be automatically generated by CAD
tools (such as the DXF format developed by AutoCAD
for enabling data interoperability). The program should
extract related information from it to identify process units
and flowsheets (Palmer and Chung, 2000). For a standard and uniform extraction of connectivity information,
XML (eXtensible Markup Language) is one possible source of information. Some CAD tools can export XML
texts, and some techniques of computer science can be
employed to capture the connectivity information from
them (Fedai and Drath, 2005; Yim et al., 2006; Thambirajah et al., 2009). This format of connectivity information
forms the basis for causality and can be converted into
matrices or SDGs essentially.
F. Yang et al.
44
2.3. SDG-based inference and model improvement.
In the safety area, fault diagnosis, especially root cause
diagnosis, and process hazard assessment, especially HAZard and OPerability analysis (HAZOP), are two different
tasks. The former is to correctly identify the root cause of
a fault. It is based on measurements and used in real-time.
The latter is used off-line to find the possible consequences due to every cause. By assuming various faults, one
can analyze possible consequences that are likely to be
triggered. Both tasks need a model and mechanism to describe how a fault propagates, and thus the SDG model
can be employed.
Based on an SDG, fault propagation can be explained
by searching for consistent paths. An arc is called consistent if the product of the sign of its source, the sign of its
destination, and the sign of the arc is equal to ‘+’. Take arc
A → B with sign ‘−’ (suppression), for example. If A is
‘+’, and B is ‘−’, then this arc is consistent, which means the fault on A can be propagated to B along this arc
because the signs show the suppression effect. A fault cannot be propagated along an inconsistent arc even though
there may be a connection. Consistent arcs are connected
successively to compose a consistent path, i.e., a reasonable fault propagation path. In Fig. 3, a consistent path is
highlighted, while there is another node with a sign ‘−’
enclosed in a circle that does not belong to any consistent
path, because the arc adjacent to it is inconsistent in spite
of the existing connection.
According to this rule of consistency, one can search
for root causes and consequences by backtracking and forward tracking, respectively. A common search algorithm
is the depth-first or width-first traversal on the graph (Iri
et al., 1979), by which the root causes or affected variables
as well as the propagation paths can be obtained.
One disadvantage of the aforementioned unit-based
method is incorrect or ambiguous inference due to the
existence of divider/header combinations or recycle loops,
because there may exist multiple paths with the same origin and terminal. This is unavoidable because of the incomplete information of qualitative SDGs. This disadvantage, however, can be partially overcome through model
improvement. Maurya et al. (2003b) analyzed model description in detail by distinguishing an initial response and
a steady-state response and modified the SDG by adding
redundant equations. Palmer and Chung (2000) developed
Fig. 3. Example of a consistent path.
the modular approach to remove redundant paths. Neither
of these efforts can solve the problem perfectly or efficiently without investigating real data, especially for large
processes. This issue will be discussed in Section 4.1.
3. Causality capture based on process data
There are many methods to capture causality information
from process data. Two of them are given below. One is
the simplest and yet a very practical one, the other is a
general but computationally complex one.
3.1. Causality capture based on cross-correlation. In
a process, variables are interacting. By analyzing the correlation between them, we can capture the causality. Although correlation does not imply causality, it can be used
as a validation. Moreover, when computing Pearson’s correlation, causes and effects can be recognized by introducing lags in a time series to find the maximum correlation.
Assume that x and y are time series of n observations
with means μx and μy and standard deviations σx and σx ,
respectively. Then the Cross-Correlation Function (CCF)
with an assumed lag k is
φxy =
E [(xi − μx )(yi+k − μy )]
.
σx σy
(6)
The expectation can be estimated by the sample CCF as
φ̂xy (k) =
⎧ n−k
⎪
⎪
(xi − μx )(yi+k − μy )
⎪
⎪
⎪
i=1
⎪
⎪
⎪
⎨
(n − k)σx σy
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
n
i=1−k
if k ≥ 0,
(xi − μx )(yi+k − μy )
(n + k)σx σy
if k < 0.
(7)
A value of the CCF is obtained by assuming a certain time delay for one of the time series. Thus the absolute maximum value can be regarded as the real crosscorrelation and the corresponding lag as the estimated time delay between these two variables. For mathematical description, one can compute the maximum and minimum values φmax = maxk {φxy (k), 0} ≥ 0 and φmin =
mink {φxy (k), 0} ≤ 0, and the corresponding arguments
k max and k min . Then the time delay from x to y is
max
k
if φmax ≥ −φmin ,
λ=
(8)
min
k
if φmax < −φmin
(corresponding to the maximum absolute value) and the
actual time delayed cross-correlation is ρ = φxy (λ) (between −1 and 1). If λ is less than zero, then it means that
the actual delay is from y to x. Thus the sign of λ provides the directionality information between x and y. The
Signed directed graph based modeling and its validation from process knowledge and process data
sign of ρ corresponds to the sign of the arc in the SDG
indicating that the correlation is positive or negative.
By this definition, ρ is a statistical estimate inevitably prone to some uncertainty due to disturbances, noise
and the size of data windows. Therefore its value should
be judged with care. Even if the two time series are uncorrelated random noises, ρ may still likely be different from
zero. Therefore the value of the CCF between two variables should be checked against a threshold. If the correlation between the two series is very weak, then the effect
of the noise will dominate the results. Therefore, in correlation analysis, only those values which are significantly
larger than a user-defined threshold (e.g., ±0.2) are considered to be evidence of correlations.
To sum up, based on estimation, the maximum CCF
is defined as the time delayed correlation coefficient (or
correlation in brief), and the corresponding argument k
is defined as an estimate of the time delay from x to y,
from which we can find the cause and the effect (Bauer
and Thornhill, 2008). Of course, correlations are based on
statistics under the assumption of linearity, and thus they
need hypothesis tests to obtain the level of significance.
Although the estimates of correlation coefficients are not
accurate, the directions are believable for most cases. Although this method is practical and easy for computation,
it has many shortcomings, some of which are explained
below.
• The nonlinear causal relationship does not necessarily show up in correlation analysis. For example,
if y equals the square of x with the time delay of
one sampling time, then, based on the time-delayed
cross-correlation, this obvious causality cannot be found because all the values are small relative to a threshold. This can be explained because the true correlation should be zero.
• Correlation simply gives us an estimate of the time
delay. The sign of the delay is an estimate of the directionality of the signal flow path. The time delay
obtained, however, is only an estimate. In addition,
the trend in a time series is ignored, and values at
different time instances are regarded as samples of
the same random event. Thus the causality obtained
by this measure is purely the time delay based on the
estimate of the covariance.
Given all the correlations between any two variables, a correlation color map can be constructed (Tangirala
et al., 2005), whose horizontal and vertical coordinates are
both variables in the same order and the color of each pixel shows the correlation between the two corresponding
variables according to a scaled color bar. This provides
an intuitive way for human beings to observe correlation,
especially for identification of similar groups of variables.
Based on these correlations with directions, one can
construct a causal network. Because correlation has the
45
property of a transitive relation, some correlations can be
explained by sequential direct causal relations. For example, the causality from A to C can be a combined result of
causal relations from A to B and from B to C. Pairwise
data analysis cannot recognize this difference without process knowledge. In addition, other related variables can
affect the correlation between two variables to lessen the
intensity, which may interfere the analysis results. In any
case, a correlation high enough, whether direct or indirect,
should be explained by an arc or a path in the corresponding SDG. This is a means of validating SDGs.
3.2. Causality capture based on transfer entropy.
From another point of view, causal relation means information transfer. Thus the measure of transfer entropy
(Schreiber, 2000) in information theory can also be employed. The transfer entropy from y to x is defined as
t(x|y) =
p(xi+h , xi , yi ) · log
p(xi+h |xi , yi )
,
p(xi+h |xi )
(9)
where p means the complete or conditional Probability
Density Function (PDF), xi = [xi , xi−τ , . . . , xi−(k−1)τ ],
yi = [yi , yi− , . . . , yi−(l−1)τ ], τ is the sampling period,
and h is the prediction horizon. The transfer entropy is a
measure of information transfer from y to x done by measuring the reduction of uncertainty while assuming predictability. It is defined as the difference between the information about a future observation of x obtained from
the simultaneous observation of past values of both x and
y, and the information about the future of x obtained
from the past values of x alone. It gives us a good sense of the causality information without having to require
the delay information. Experience suggests that one takes
τ = h ≤ 4, k = 0, and l = 1 for the initial trial, while the
usual way is to test and compare various results from different values of τ and h. If the transfer entropies in two directions are considered, then t(x → y) = t(y|x) − t(x|y)
is used as a causality measure to decide the quantity and
direction of information transfer (Bauer et al., 2007).
In Eqn. (9), the PDF can be estimated from a histogram or kernel methods (Silverman, 1986), which are
nonparametric methods, to fit any shape of the distributions. Here the Gaussian kernel method is used because
it is more robust than the naive histogram-based method.
The Gaussian kernel function is defined as
1 2
1
K(v) = √ e− 2 v .
2π
(10)
Thus a univariate PDF can be estimated by
N
1 p̂(x) =
K
N h i=1
x − xi
h
,
(11)
where N is the number of samples, and h is the bandwidth chosen to minimize the mean square error of the
F. Yang et al.
46
PDF estimation calculated by h = c · σ · N −1/5 , where
c = (4/3)1/5 ≈ 1.06 according to a ‘normal reference
rule-of-thumb’ (Li and Racine, 2007).
For the multivariate case (q dimensional), the estimation of PDF is
p̂(x1 , x2 , . . . , xq )
1
=
N h1 · · · hq
N
x1 − xi1
xq − xiq
·
K
···K
,
h1
hq
i=1
where
y(i)+0.5y(i-1)=x(i-5)
x
y
x(i)=1-0.5y(i-1)
(a)
(12)
−1/(4+q)
hs = cσ̇(xis )N
i=1 · N
for s = 1, . . . , q, and the notation is the same as the univariate case.
Compared to the approach based on crosscorrelation, this one can be applied to more general
conditions such as nonlinear relations. In the above
(nonlinear) example, the causality cannot be validated
based on the cross-correlation. Given a lag of 1, however,
the transfer entropies from x to y and vice versa are 0.27
and 0.01, respectively, and thus the causality from x to
y can be confirmed, which is consistent with the actual
setting.
Transfer entropy shows the information transfer in
each direction. Thus this method provides more insight
into causality for complex systems especially for the case
with recycles. In Fig. 4(a), x and y are connected directly via a forward path and a recycle path. In the forward
channel from x to y, the relation is via an AR model, i.e.,
y(i) + 0.5y(i − 1) = x(i − 5), whereas there also exists a
feedback channel from y to x, i.e., x(i) = 1 − 0.5y(i − 1).
Thus the information transfer lies in the two channels. If
one is only concerned with the causality measure, then
t(x → y) (in Fig. 4(b)) indicates that the causality is from
x to y. However, if transfer entropies t(y|x) (in Fig. 4(c))
and t(x|y) (in Fig. 4(d)) are studied, then both arcs can be
validated.
To sum up, this method is more general than the previous one and is able to reveal the essential relationship
even when the lag is inexplicit. There are several parameters to be set by users, which provide some freedom. It
is, however, highly dependent on the estimation of PDFs
(although it can have any non-Gaussian forms). Thus the
computational burden is higher. However, when using this
method, the time delay cannot be estimated, and the arc
signs in SDGs cannot be obtained. In real applications,
one may mainly choose correlation analysis for validation
but sometimes the transfer entropy can bring in additional
insights.
There are other alternative methods to capture the
causality between time series such as Granger causality
and predictability improvement (Lungarella et al., 2007).
(b)
(c)
(d)
Fig. 4. Transfer entropy measures: two variables and their relations (a), transfer entropy from x to y (b), transfer entropy from y to x (c), causality measure which is the
difference (d).
Each has its own advantages and limitations. They complement each other and no one method is powerful enough
to replace the others. Hence data analysis is a combination
of various methods.
Signed directed graph based modeling and its validation from process knowledge and process data
4. Mutual validation of process description
Process knowledge and process data are two means to capture causality information in a process. Neither, however,
is sufficient for practical use because of redundancy and
errors in the resulting models. We should combine them
by mutual validation.
4.1. Using process data to validate knowledge description. A causal network constructed based on process
knowledge is essentially closer to the real causality. Ideally, it covers all the possible paths. It also has the ability
to avoid indirect relations by avoiding parallel paths. For
example, if A can reach B and B can reach C, then A
can reach C. However, if the reachable path from A to C
is just realized by the serial connection of the two paths
from A to B and from B to C, then the direct arc from
A to C can be deleted, otherwise there should exist sound reasons to include different parallel paths to achieve
the reachability from A to C, resulting in a necessary arc.
The major problem of this causal network constructed by
process knowledge is the lack of quantitative information
to confirm its reliability. In fact, there are usually redundant and irrelevant arcs that should be deleted.
The reasons for this redundancy include many cases
whereas the essential reason is that connectivity is merely
a necessary but insufficient condition for causality. While
detailed information can also be obtained to exclude the
inexistent arcs or choose the dominant path, the effort would be multiplied because quantitative process knowledge
is needed in addition to qualitative information. In the modular approach (Palmer and Chung, 2000), this task is left
to the user, although semi-automated; it still highly relies
on experience. Thus we limit the process knowledge of
concern to P&IDs, that is to say, qualitative connectivity information. Under this circumstance, complex structures, such as dividers, headers, or recycle loops, often make the qualitative algebra provide ambiguous results that
are difficult to improve according to qualitative information because the intensity of each arc is unknown. This is
an intrinsic problem of the SDG model, although a few
researchers have made some improvements (Palmer and
Chung, 2000; Maurya et al., 2003b).
In general, it is difficult to exclude physically broken
(e.g., valve block), behaviourally nondescript loop (e.g.,
control loop), or extremely weak (due to attenuation of
signals) paths. To validate the causal network, one should
resort to process data for quantitative evidence. If there is
no data support for reachability, then the causality should
be excluded. For the tank system shown in Fig. 1, if the
data of F1 , F2 , L, and K (take controller output as an
alternative) are available and sufficiently excited, then the
arcs in Fig. 2(c) can be validated except F2 → L because
the control determines the value of L.
47
4.2. Using process knowledge to validate data-based
relations. Although there are several data-based methods to capture causality, they are developed to find the
causality between two variables. Real systems, however,
are multivariate, the causality within which is shown as
a network with weights based on the causality measures
between every two variables. Thus a causality matrix is
obtained to reflect the magnitude of the causality of each
pair of variables, and the direction is determined by the
time delay in correlation analysis or the sign of measure
in information transfer computation. The topology of the
causal network, however, also relies on the propagation
relations by screening the indirect relations. According to
Bauer and Thornhill (2008), one of the two typical topologies (extreme cases) is generated according to the number of nonzero entries in the first row and above the main
diagonal of the causality matrix, while the real topology
is the combination of these two topology forms. With the
aid of a correlation test and a directionality test, one can
select the evident relations first and construct the network
(Yang et al., 2010b). A method based on correlation is
proposed, which introduces the resulting quantitative information obtained by CCF computations.
• Step 1: In matrix P, select the maximum value in the
elements that has not been used and tested.
• Step 2: Check the results of the correlation test and
directionality test. If the correlation value fails to
pass the tests, then stop.
• Step 3: Check the result of the consistency test for all
the variables in the existing arcs. If it fails, then go to
Step 5.
• Step 4: Add an arc corresponding to this element with
an estimated time delay. The sign of the arc is determined by the sign of the element.
• Step 5: Go to Step 1.
Nevertheless, the resulting network can be incorrect and
may be inconcise without validation by process knowledge.
One possible way to validate an acceptable causal network is to check the reachability between the two variables for evident causality; this can be realized by searching
consistent paths. This treatment, however, cannot exclude
indirect relations; the arcs are selected according to their
magnitudes of causality. For example, if causal relations
exist from A to B and from B to C, there may be strong
causality between A and C that can be confirmed by reachability, yet it is actually redundant. The study to identify whether the path is direct or indirect is underway in the
temporal as well as the spectral domain (Gigi and Tangirala, 2010). Although it is possible, it is not efficient without looking at process knowledge. Thus, in most cases,
F. Yang et al.
48
one should make good use of process knowledge in the
modeling procedure.
For the tank system as shown in Fig. 1, based on the
data of F1 , F2 , L, and K, one can easily obtain the causal
relations from F1 to F2 , from K to F2 , and from F1 to K
when the level control is in effect. These arcs can be validated by reachability check. If the data set also includes
a transient response, then other causal relations, such as
from F1 to L, can be detected. The SDG can be constructed accordingly and validated. The structure related to L is
slightly different from Fig. 2(c) because of the level control.
the P&ID, they are found to be associated with the level
and density controls in the same pump box.
One can focus on the cross-correlations and time delays between the six variables for further analysis. They
are shown by the following correlation matrix P (comprising all the correlation coefficients between two variables)
and the causality matrix Λ (comprising all the estimated
time delay λs from one variable to another).
Table 1. Key process variables in the FTPH process: Line A for
causality analysis.
5. Industrial case study
Notation
x1
x2
x3
x4
x5
x6
Consider a real industrial system, the Final Tailings Pump
House (FTPH) process at Suncor Energy Inc. in Fort
McMurray, Alberta, Canada.
5.1. Process description. The flow sheet for this process is shown in Fig. 5, where some texts have been made illegible and the control strategy is omitted for confidentiality reasons. The tailings from upstream are pumped into a distributor and then processed in parallel cyclopacks and pump boxes, and finally discharged into the
ponds. There are five parallel lines from the cyclo-pack
downwards, where Lines A/B/C/E are structurally identical while Line D is distinct. Based on the pressure of the
distributor, a prioritization program is implemented on the
parallel lines and Line A is therefore the most important.
The single-loop, cascade, and selective control strategy are applied, including distributor pressure control,
cyclo-pack pressure and underflow control by adjusting
the number of cyclones opened, gypsum addition flow rate control, pump box level control and discharge density control by adjusting Cold Process Water (CPW), pump
box level control by adjusting pump speed, pump box discharge flow rate control by adjusting pump speed, and
Mature Final Tailing (MFT) addition flow rate control. For
this process an SDG constructed from process knowledge
is shown in Fig. 6.
For simplicity, six key variables xi (i = 1, . . . , 6) in
Line A are chosen, which are y10, y16, y18, y21, y28,
and y30, respectively, as indicated in Table 1. The corresponding subsection of the SDG (Fig. 7(a)) is extracted
from the SDG in Fig. 6. This can be easily done by checking reachability in the SDG model.
x4
x2
x6
x5
Tag name
y10
y16
y18
y21
y28
y30
Description
distributor pressure
gypsum addition flow rate
gypsum density
sludge header pressure
sump density
sump level
x1
x3
(a)
(b)
(c)
5.2. Using process data to validate knowledge description. Based on the process data of the variables in Line A over a one-week long period (with one minute sampling frequency), the correlation color map obtained is displayed in Fig. 7(c). We use these data and corresponding
time delays for SDG validation. For example, the bottom
left corner is a cluster of correlated variables. By checking
Fig. 7. SDG and its validation by cross-correlation: subsection
of the SDG concerning six important variables in which
the thick lines are validated (a), correlation color map of
the six important variables (b), correlation color map of
all the variables in Line A (c). Readers can contact the
authors ([email protected]) for a colored version of this figure for better visualization.
Signed directed graph based modeling and its validation from process knowledge and process data
49
Fig. 5. Partial flowsheet of the FTPH process (texts made illegible for confidentiality reasons).
y1
y2
y6
Line prioritization disabled
xy 101
y3
y7
y4
y8
y 12
Line prioritization enabled
y 11
PR
1.1
0
y21
y9
y5
y 14
x4
HS
1.2
y11
y 31
HS
HS
4.1
y 22
y 23
y 30
x6 4.1
y 33
4.2
4.2
.
3.1
5
y 27
y 26
5
3.1
y 16
y 17
y 25
y 32
HS
HS
2
3.2
Line D
y 29
2
x2
y 24
3.2
Controls:
0
distributor pressure control
1.1 cyclo-pack pressure control (by adjusting no. of cyclones opened)
1.2 cyclo-pack underflow control (by adjusting no. of cyclones opened)
2
gypsum addition flowrate control
3.1 pump box level control (by adjusting CPW)
3.2 pump box discharge density control (by adjusting CPW)
4.1 pump box level control (by adjusting pump speed)
4.2 pump box discharge flowrate control (by adjusting pump speed)
5
MFT addition flowrate control
y 28
x5
x3
y 18/ 19/ 20
Line A/B/C/E
Fig. 6. SDG of the FTPH process. Relationships between variables are shown as thick lines while control signal paths are shown as
thin lines. Solid and dashed lines show positive (reinforcement) and negative (reduction) causal relations, respectively. A dotted
line is a fault propagation path.
F. Yang et al.
50
⎡
⎢
⎢
⎢
P=⎢
⎢
⎢
⎣
⎡
⎢
⎢
⎢
Λ=⎢
⎢
⎢
⎣
1
0
⎤
.28 −.28 −.18 .39
.41
1
.36 −.31 .74
.50 ⎥
⎥
1
−.14 .10 −.11 ⎥
⎥ , (13)
1
−.25 −.24 ⎥
⎥
1
.75 ⎦
1
⎤
−271 220
32
49
5
0
−360
2
1
1 ⎥
⎥
0
−357 359 −360 ⎥
⎥ . (14)
0
20
60 ⎥
⎥
0
−1 ⎦
0
For example, the (i, j)-th elements of P and Λ are the
correlation and the estimated time delay between variables i and j, respectively. Due to the symmetric and antisymmetric properties of these two matrices, only the elements above the diagonal are computed and shown here.
Rather than looking at the correlation matrix in the numerical form, it is better to look at the color coded correlation matrix as shown in Fig. 7(b) which is a portion of
Fig. 7(c).
In Eqn. (13), those values exceeding a pre-set threshold (such as 0.4) can be used to validate some of the
arcs in the subsection of the SDG shown in Fig. 7(a) as
thick lines. For example, correlation from x5 to x6 is 0.75
and the time delay is −1, so the arc from x6 to x5 is validated. Similarly, the arcs from x2 to x5 , x2 to x6 , x1 to x6 ,
and x1 to x5 are also validated. Note that very large time
delays do not make sense because they show the ineffectiveness of this measure and the computed correlation is
considered invalid.
There are still some arcs that have not been validated. Thus we resort to a more general but more complex
measure—transfer entropy. To obtain a rough insight, if
we look at the time trends, then we observe that the trends
of x2 and x4 do not look like the other four variables.
This shows that, during this period, the relations associated with x2 and x4 are not strong enough to be validated.
We have to examine more data with sufficient excitement
to check if there is a discernible relation between these
variables. However, the trends of the other four variables
have apparent similarities that should be definitely due to
causality. In order to reduce the computational load, we
extract from the previous data set only 200 minute records
of continuous data shown in Fig. 8(a).
The transfer entropy measure is used to compute the
information transfer between these four variables where τ
is assumed to lie between 1 and 10. When τ is 9, the transfer entropies from x1 to x6 , from x3 to x5 , and from x5 to
x6 all reach their individual maximum values: 2.01, 1.48,
and 1.41, respectively. Thus the bidirectional relationship
between x5 and x6 is validated, and the reachability from
x3 to x5 is also validated. They are marked in Fig. 8(b) as
thick lines.
By combing the above two methods, we found that
most of the arcs have been validated except from x4 to
x6 and from x1 to x5 . The former can be explained by
process knowledge because x6 is the sump level, affected
by quite a few variables due to various feeds. To validate
this, we have to extract the influence of each variable by
signal conditioning, which is an ongoing work in the framework of multivariate systems. The latter is an indirect
relation, i.e., x1 and x5 are related with the relay of x6 .
Since quantitative information is missing in the SDG model, the transitive property may be weakened due to the attenuation during the propagation. The time-delayed crosscorrelations from x1 to x6 and from x6 to x5 are 0.41 and
0.75, respectively, but from x1 to x5 it is 0.39, less than the
above two. Moreover, the time delays of the above two are
5 and 1, respectively, while the indirect one is 49, making
the validation result of this arc unacceptable. Therefore,
data analysis is usually used to only validate direct arcs.
5.3. Using process knowledge to validate data-based
relations. Starting from data-based methods, for examTime Trends
4
3
2
1
1
200
Samples
x4
x2
(a)
x6
x5
x1
x3
(b)
Fig. 8. Time trends and validation results by transfer entropy:
time trends of four important variables (1: x1 ; 2: x3 ; 3:
x5 ; and 4: x6 ) in the process (a), SDG and validated arcs
(thick lines) (b).
Signed directed graph based modeling and its validation from process knowledge and process data
ple, correlation analysis, one can obtain the causal network. According to the procedure in Section 4.2 and
Eqn. (13), 0.74 is selected, so there is an arc between x5
and x6 , and the sign is ‘+’. The corresponding element in
Λ is −1 as shown in Eqn. (14). Thus the direction is from
x6 to x5 . Similarly, the second arc is x2 → x5 , and the
third one is x2 → x6 . For the latter, because the two associated nodes have been used, the consistency test should
be undertaken. The time delays associated with the three
arcs are 1s that are reliable. Other arcs are built one by
one. Note that arc x1 → x5 is ignored because the time
delay is 49, which fails to pass the consistency test. The
obtained SDG is shown in Fig. 9.
However, from reachability analysis based on the
SDG, arc x2 → x4 does not make sense and should be
deleted because there is no path connecting them. From
process knowledge, they are parameters on different pipes, and in the P&ID of Fig. 5, they are independent. One
explanation for this link is that there is another cause or
upstream unit resulting in the changes in both of them.
This incorrect arc may bring out wrong results if the root
cause is x2 because x4 does not depend on it.
5.4. Application of SDGs in fault propagation analysis. Given a validated SDG, fault propagation can be
analyzed qualitatively based on consistent paths. This is
important in any conclusive significant HAZOP analysis
and also in fault detection and isolation, especially root
cause analysis, where SDGs can help.
In this case, a fault propagation path is shown by the
dotted lines in Fig. 6 meaning that the pressure change
in the distributor (x1 ) can affect the parameters in cyclopacks and pump boxes (x5 and x6 ) in turn. During the HAZOP study, all the consistent paths should be considered
and the corresponding consequences should be evaluated.
If the domain of influence of one variable is large, the intensity of influence is strong, or the consequence is severe,
then some appropriate measures should be taken. In this
case study, x1 is important because it has wide influence
on almost all downstream variables. Thus the controller on
it is well tuned and the line prioritization is implemented
51
to lessen the risk. On the other hand, root cause analysis
is undertaken online. When the variables on this path are
showing disturbances, then one can trace immediately the
starting point and that may be identified as the root cause of fault propagation. The automation of this procedure
will help operators quickly identify the symptom of the
abnormal situation.
6. Concluding remarks
In this paper, modeling and fault propagation analysis based on SDGs are proposed for application to large-scale
industrial systems. The SDG has its advantage of simplicity and good matching with process data. It is feasible
even when no model is available. Thus it has great potential in many process analysis applications. Although both
process knowledge and process data can be used to capture the causality between variables, neither of them can be
solely used to find the true causality without validation.
For the validation of SDGs modeled by process knowledge, especially by P&IDs, two typical
temporal-domain data-based methods are described.
Cross-correlation is relatively easy to compute and thus
very practical for determining signs and estimating time
delays. As a more general but more costly computational measure, transfer entropy gives a complementary validation from the information transfer perspective. In both
methods, arcs can be validated one by one, as matrices P
and Λ are not necessary, and hence the total computational effort is proportional to the number of arcs. The above
two points of view are illustrated in this paper to show
the fusion of two data sources in process modeling. Based
on the validated model, fault propagation analysis can be
performed.
Sometimes, process data are easy to obtain while process knowledge is insufficient. For this case, data-based
methods can be applied first to build a causal network,
and then reachability check can be applied to validate this
network by process knowledge. Moreover, if the causal
network is not important whereas only some causality relations are of concern, then this method is more useful.
Acknowledgment
x4
x5
1
2
1
x2
1
x6
5
x1
Fig. 9. Causal network obtained by correlation analysis. Solid
and dashed lines are positive and negative correlations,
respectively. Numbers are the estimated time delays.
The authors are grateful for the financial aid from NSFC
(60736026 and 60904044), NSERC SPG and IRC at the
University of Alberta and the Tsinghua National Laboratory for Information Science and Technology (TNList)
Cross-discipline Foundation. Data access from Dr. Ramesh Kadali at Suncor Energy Inc. is gratefully acknowledged.
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Fan Yang earned a B.Eng. degree in 2002 and a
Ph.D. degree in 2008, both from Tsinghua University. From 2009 to 2011 he worked as a visiting professor and PDF at the University of Alberta. Currently he is a lecturer at Tsinghua University. He obtained the Young Research Paper
Award from the IEEE CSS Beijing Chapter in
2006. His research interests include topology modeling of large-scale processes, abnormal events
monitoring, process hazard analysis, and smart
alarm management.
Sirish L. Shah has been with the University of Alberta since 1978, where he currently
holds the NSERC-Matrikon-Suncor-iCORE Senior Industrial Research Chair in Computer Process Control. He was the recipient of the Albright
& Wilson Americas Award of the Canadian Society for Chemical Engineering (CSChE) in recognition of distinguished contributions to chemical engineering in 1989, the Killam Professor in
2003, and the D.G. Fisher Award of the CSChE
for significant contributions in the field of systems and control. The main
area of his current research is fault detection and diagnosis, alarm design
and analysis, process and performance monitoring, system identification
as well as design and implementation of soft sensors.
53
Deyun Xiao is a professor in the Department
of Automation at Tsinghua University. Since his
graduation from Tsinghua University in 1970, he
has been with this department as a faculty member. His research interests cover a wide range of
topics related to the area of control science and
engineering, including fault diagnosis and hazard assessment, system identification and parameter estimation, multi-sensor data fusion, intelligent measurement and control, intelligent sensing techniques, computer control systems, real-time database systems,
industrial monitoring software, enterprise comprehensive automation
systems, and intelligent transportation systems.
Received: 18 January 2011
Revised: 20 July 2011